Using pivot signs to reach an inclusive definition of
rectangles and squares
Maria Bartolini Bussi, Anna Baccaglini-Frank
To cite this version:
Maria Bartolini Bussi, Anna Baccaglini-Frank. Using pivot signs to reach an inclusive definition
of rectangles and squares. Konrad Krainer; Naďa Vondrová. CERME 9 - Ninth Congress of the
European Society for Research in Mathematics Education, Feb 2015, Prague, Czech Republic.
pp.1891-1897, Proceedings of the Ninth Congress of the European Society for Research in
Mathematics Education. <hal-01288449>
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Using pivot signs to reach an inclusive
definition of rectangles and squares
Maria G. Bartolini Bussi and Anna Baccaglini-Frank
Università di Modena e Reggio Emilia, Dipartimento di Educazione e Scienze Umane, Modena, Italia, [email protected]
We present some fragments of a teaching experiment
realized in a first grade classroom, to sow the seeds for
a mathematical definition of rectangles that includes
squares. Within the paradigm of semiotic mediation, we
studied the emergence of pivot signs, which were exploited by the teacher to pave the way towards an inclusive
definition of rectangles and squares. This was done to
favor overcoming children’s spontaneous distinction
of these figures into distinct categories, reinforced by
everyday language. The experiment is an example of an
approach towards the theoretical dimension of mathematics in early childhood.
Keywords: Bee-bot, first grade, pivot signs, rectangles,
squares.
INTRODUCTION
Rectangles and squares represent a paradigmatic example of the conflict between the perceptual experience and the theoretical needs of a mathematical definition (on this persisting conflict also see Hershkowitz,
1990; Clements, 2004; Fujita, 2012; Koleza & Giannisi,
2013), where squares are to be considered as particular rectangles (we will refer to a definition of rectangles that includes squares as being inclusive). Mariotti
and Fischbein (1997) claim that “from the figural point
of view squares and non-square rectangles look so
different that they impose the need of being distinguished at least as much as triangles and quadrilaterals” (Mariotti & Fischbein, 1997, p. 224). Actually
the difficulty of naming and classifying geometrical
figures (and, in particular, squares and rectangles),
according to inclusive criteria, seems to depend on
different reasons:
――
the implicit constraints of everyday language:
for instance, both in Italian and in English (as
well as in other European languages) the names
CERME9 (2015) – TWG13
“quadrato” [square] and “rettangolo” [rectangle]
hint at a complete separation of the figures into
two different classes (square and not-square rectangles);
――
some widespread improper practices in school
which reinforce the separation between squares
and rectangles (for instance, activities with attribute blocks, where squares and non-square
rectangles are classified in different sets).
Hence, teaching needs to orient learning towards
an inclusive definition. The question is: at what age?
We claim that, although this choice may create a discontinuity between everyday language and school
language, it is possible from early childhood to sow
the seeds of an inclusive definition, focusing on
the experience of walking along or drawing a rectangular path, where the change of direction in the
four angle vertexes has the potential to attract the
students’ attention. In the following, we report on
some fragments of a long term teaching experiment,
carried out within the theoretical framework of semiotic mediation (Bartolini Bussi & Mariotti, 2008).
Additional details are discussed by Bartolini Bussi
and Baccaglini-Frank (2015).
THEORETICAL FRAMEWORK
In order to design and to analyze the teacher’s role in
the classroom teaching process, we adopted the theoretical framework of semiotic mediation (Bartolini
Bussi & Mariotti, 2008; Bartolini Bussi, 2013). The
design process is represented by the reciprocal relationships between the tasks, the artifact, and the
mathematical knowledge at stake. In this relationship
the semiotic potential of the artifact is made explicit.
The artifact is the bee-bot, a small programmable robot represented in Figure 1 (also see the next section).
When children are assigned a task they engage in a
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Using pivot signs to reach an inclusive definition of rectangles and squares (Maria G. Bartolini Bussi and Anna Baccaglini-Frank)
rich and complex semiotic activity, producing traces
(gestures, drawings, oral descriptions and so on), that
we refer to as “situated texts”. The teacher’s job is to
collect all these traces (by observing and listening
to the children), to analyze them and to organize a
path for their development into “mathematical texts”
that can be put in relationship with the fragments of
mathematics knowledge that are to come into play.
The process of semiotic mediation also concerns the
functioning of semiotic mediation within the classroom. The teacher acts as a cultural mediator, in order
to exploit, for all students, the semiotic potential of
the artifact (the bee-bot in our case). In this last process, Bartolini Bussi and Mariotti (2008) identify three
main categories of signs: artifact signs, pivot signs,
and mathematical signs. Artifact signs “refer to the
context of the use of the artifact, very often referring
to one of its parts and/or to the action accomplished
with it. […]”; mathematics signs “refer to the mathematics context” and pivot signs, which “refer to specific
instrumented actions, but also to natural language,
and to the mathematical domain” (ibid, p. 757). Pivot
signs can be particularly useful for fostering a transition from situated “texts” to mathematical texts. Pivot
signs develop and are enriched by their relationships
with other pivot signs, hence building a network of pivot signs. Mathematical signs are not intended to suddenly substitute artifact signs; in fact the latter may
survive for some time, especially for lower achievers
or in cases in which the formal mathematical definition and the reasoning of the corresponding concepts
require long term processes to be achieved.
Within this framework, our study addressed the following research questions:
(1): How might a long-term process of semiotic mediation
that exploits the semiotic potential of the bee-bot with
respect to the development of an inclusive definition of
rectangles look for first graders?
(2): In particular, which kind of pivot signs (if any) can
be identified and exploited during such long-term process?
THE CHOSEN ARTIFACT: THE BEE-BOT
The bee-bot (Figure 1) is a small programmable robot,
especially designed for young students. Its ancestor is
the classical LOGO turtle, originally a robotic creature
Figure 1: Bee-bot’s back
that could be programmed through an external computer to move around on the floor (LOGO Foundation,
2000). It is not necessary to have any additional computer to program the bee-bot; this can be done simply pressing a sequence of command buttons on its
back. When the programme is executed, the bee-bot
moves on the floor: the execution of each command
is followed by a blink of the eyes and by a short beepsound. The bee-bot hints at many sets of meanings
and mathematical processes, partly related to mathematics and partly related to computer science, for
instance: counting (the commands); measuring (the
length of the path, the distance); exploring space, constructing frames of reference, coordinating spatial
perspectives, programming, planning and debugging.
In a long term teaching experiment, all these sets of
meanings are at stake, sometimes in the foreground
and sometimes in the background. Focusing on any
set of them depends on the adult’s teaching intention.
The bee-bot walks on the floor and traces paths that
can be perceived, observed, described with words, gestures, drawings, sequences of command-icons and so
on. Paths (either traced or imaginary, when no trace
mark is actually left) constitute a large experiential
base to “study” some plane figures, that can be traced
using the available commands. These are polygons
with sides measured by a whole number of steps and
with right angles only. With the additional constraint
of being convex, the bee-bot can be programmed only
to turn “left” or “right” (with respect to itself ), and
therefore the convex polygons it can trace are always
rectangles (including squares). Moreover, in experiences where “pretending to be the bee-bot” is essential,
children embrace the robot’s perspective: they move
with the bee-bot and they see through its eyes. In particular, when walking along a closed convex path and
ending up where they started, the children turn 360o
in four equal “chunks” during which their orientation
is perceived as essential (they find it important to end
up facing the same direction as when they started).
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Using pivot signs to reach an inclusive definition of rectangles and squares (Maria G. Bartolini Bussi and Anna Baccaglini-Frank)
THE TEACHING EXPERIMENT
Above we have discussed some features that define
bee-bot’s high semiotic potential with respect to the
emergence of an inclusive definition of rectangles,
characterized by the property of four right angles.
Our teaching experiment was designed to capitalize
on bee-bot’s potential of fostering awareness of the
“four right angles” property of generic rectangles (including squares).
left hand (for a left turn) up and to the right or left in
a rotation (turning gesture). The turning gesture was
mirrored by the student-teacher, as a pivot sign with
respect to the notion “angle” in a path.
Pretending to be a bee-bot
During this session the students were asked to work
in pairs: one pretended to be the bee-bot and the other
gave the first commands to move according to some
undisclosed (to the first student) path. The intention
was to guide the children to focus their attention
Several sessions (15) were carried out in a first grade on the turn command. Typical words used were be
classroom at the beginning of the school year, for “Straight Ahead” “Left” “Right” “Backwards” usually
4 months (more or less once a week) either in the without quantifying the number of steps, and freclassroom or in the gym, with a careful alternation quently combining a translation with a change of diof whole class or small group activity (with adult’s rection (rotation). For example, when a student said
guidance) and some individual activity. Each session “left” the bee-bot student frequently would not only
was carefully observed by the teacher, by a student turn left, but s/he would also take a step in that directeacher or by a researcher (the second author of this tion, or even just take a step to his/her left without
paper), with the collection of students’ protocols, pho- even turning in that direction. The student-teacher’s
tos, and videos. The tasks were designed by the whole intervention here was fundamental in focusing the
research team, drawing on the initial intention and children’s attention on “turn” commands, which led
on some changes implemented “on the fly” based on to their beginning to explicitly consider rotations as
episodes that occurred during the experiment. Due important elements per se, without having to associto space constraints it is not possible to report on all ate them to steps.
the details, so we have focused on particular sessions
where the production of signs was very rich and fun- Constructing paths
damental for preparing the final summary texts and Several activities were designed around tracing difposter for the students (see Figure 6 in this paper, and ferent kind of paths on the floor. When the aim was
Bartolini Bussi & Baccaglini-Frank, 2015).
to produce particular letters of the alphabet, the students’ attention was focused mostly on the “possible”
Observing programmed bee-bots
and “impossible” letters: they empirically discovered
In this session, students were given two bee-bots that that some capital letters (e.g., L, T. I) could be traced
had ahead of time been programmed with the same out, whilst others could not (e.g., B, A, D, O). In fact,
sequence. The task was: Describe what they do. The neither acute angles (“sharp points”) nor circular arcs
students watched the twin bee-bots move together, (“fat curves”) could be traced by the bee-bot. Children
starting facing in the same or in different directions, produced many examples of combinations of words,
and then moving separately. Then the memory of one gestures and drawings, aiming at distinguishing the
of the bee-bots was erased (CLEAR command-icon) shapes (letters) which could or could not be drawn.
and the students were asked to reprogram it so that it There was a particularly rich production of words
would move just like the other bee-bot. The students’ such as “angles”, “(fat) curves”, “diagonals”, “(sharp)
productions concerned both global and local aspects. tips/points”, “broken lines” and of related gestures
Global aspects refer to the perception of a path as a and drawings. Suddenly, within this experience, an
whole (as if bee-bot had drawn it on the floor), whilst important event took place; this will be the seed of an
local aspects refer to special points of the path. An inclusive definition of rectangles.
example of the former is the expression “it did an L”;
an example of the latter is “they switched the turn”. The main pivot sign: the “squarized” O
Both aspects also appeared in gesturing: the path is In a small group the following exchange occurred:
represented by a single pointer finger tracing a path
in the air (tracing gesture), whilst turning is repreStudent-teacher: …Did you do an O?
sented by moving the right hand (for a right turn) or
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Using pivot signs to reach an inclusive definition of rectangles and squares (Maria G. Bartolini Bussi and Anna Baccaglini-Frank)
Student: No. Then it could do like this this this
and this [he gestures four consecutive
right angles] a squarized O. Ah, then it
can make a square!
We have translated a non-existing Italian word
(quadratizzato) into a non-existing English word
(squarized). Other students started talking about
“squarized Os” and other possible “squarized letters”,
intending letters that include one or more squarized
Os within them (e.g., P, B). These squarized 0s were
acknowledged by the teacher and the research team
as pivot signs, hinting at both the perceived path produced by the bee-bot (artifact sign) and at a square (a
figure, interpreted as a mathematical sign). The importance of the four consecutive right angles suggested to orient children’s attention towards this feature,
that seemed to put in shade the length of each piece
of the traced path (the sides) and to put in the foreground the four changes of direction, common to all
squarized Os.
Focusing on the four right angles
In the students’ complex experience, each right angle
appeared with seemingly different meanings, that
also affected the signs used. These, initially, were
mainly dynamic and related either to the student
pretending to be a bee-bot or to the bee-bot:
would make if a marker were used”, she chose to mirror a sign produced by a student “a turn like this” close
to the turning point of the path (see Figure 2).
The sign had the potential to become a pivot sign with
respect to the notion of “angle” (external angle): it recalls the command-icon on bee-bot’s back, but it is
somewhat decontextualized, since there appears to
be no explicit mention to the bee-bot
In addition to these dynamic signs, as the teaching
experiment went on, the children developed other
signs, which lacked such dynamic components:
c) Hands-meeting gesture referring to the point in
the path traced by the bee-bot;
d) Gestures to interpret a static figure (referring
to a dynamic experience);
e) Verbal utterance of the list of commands (uttered during or after the programming of the
bee-bot);
f) List of commands written horizontally.
a) Dynamic change of direction of the student
pretending to be a bee-bot;
b) Dynamic change of direction of the bee-bot under the effect of the turn command.
In both these cases, however, the angle was the external angle, i.e. the region swept by the gaze of either
the student or the bee-bot while changing direction.
When the researcher proposed to draw the paths in
a “faster way: using a mark like the one the bee-bot
Figure 3: The students’ gesture
First we describe the hands-meeting gesture (type c).
While exploring figures that represented rectangles,
including squares, a powerful gesture was realized
by one of the groups of children and rapidly imitated
by others: the two hands coming together at a right
angle (Figure 3). The gesture emerged as the students
tried to explain the property that all squarized O’s (be
they “allungati” [stretched] or “perfetti” [perfect]) had
in common: all the four angles (internal angles) are
equal and right. Moreover the gesture stresses the
vertex as an important feature of the angle. Signs of
type d were identified, for example, in the argument
presented below (Figure 4), on how the angles of a
square or rectangle have to be (as opposed to angles
such as the ones of the parallelogram that was included in one of the worksheets).
Figure 2: Sign for the right angle
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Using pivot signs to reach an inclusive definition of rectangles and squares (Maria G. Bartolini Bussi and Anna Baccaglini-Frank)
Figure 4: Veronica’s gestures
Veronica: “[in a square or rectangle] the angles go
forward, turn right. The segments have
down straight…[in the parallelogram]
to all be equal.
they are a bit down to the right and a
bit down to the left. It has to go straight, Type f emerged in activities in which children had
not like this and down, it shouldn’t be learned to represent traced paths as written sequenca bit down like this one [she moves her es of commands, typically in a horizontal line, from
pencil in the air along a slanted line left to right. Within these sequences they searched
with respect to a horizontal bottom for regularities allowing them to distinguish differline]. Instead it has to go straight like ent types of “squarized Os”. Figure 5 shows signs left
this and like this…it has to be straight on the interactive white board after a discussion on
like the line but a bit lying down [she “stretched squarized Os” (non-square rectangles) with
marks the lower horizontal line].”
respect to “perfect squarized Os” (squares). We note
here how some students’ language (in this and othSigns of type e appeared when the students’ attention er occasions) seemed to be evolving into condensed
was drawn to the “length of the path”. Sometimes the pre-algebraic forms, such as a+b+a+b, that could eventurn command was in shade, as it did not lengthen tually become expressions like 2a+2b for the rectangle
the path perceived while the bee-bot spun around. and 4a for the square (a particular case in which a=b).
However the number of commands for paths with an- In this teaching experiment, however, we did not pick
gles, was not the same as the number of steps forward. up on these expressions, leaving them only as little
So sometimes the turn command was still skipped
germs to be nurtured by the teacher in future years
(children 1, 2, 3, below). While in some of the children’s (perhaps even during the second grade).
utterances it was acknowledged (child 4, below) as a
command like the others (it is represented by a similar Focus on the shapes as wholes
button and it is executed with by a beep and a blink Shapes as wholes were focused on from the very
of bee-bot’s eyes).
beginning of the teaching experiment, with either
verbal descriptions alone or also with hand gestures.
Child 1: Three steps then three then three then After the introduction of the idea of squarized Os, the
three we make a square, because it is the adults involved in the experiment started mirroring
same ends, the same length.
students’ utterances involving the words “rectangles”
Child 2: Instead, the other one has 1, 2-1, 2, 3-1,
2-1, 2, 3, it has two the same and two the
same.
Child 3:
The other was three, two, three, two. Not
all equal.
In contrast
Child 4: Two forward, turn right, two forward,
turn right, two forward, turn right, two
Figure 5: Agreed-upon signs for the programmed sequences and
the paths
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Using pivot signs to reach an inclusive definition of rectangles and squares (Maria G. Bartolini Bussi and Anna Baccaglini-Frank)
and “squares”. As expected, when the attention was not
brought to the word squarized O students spontaneously tended to partition the two situations, implying that
“rectangles” had pairs of sides with different lengths
(“equal in front of each other”) while “squares” had
sides that were “all equal”. For some children this property seemed to persist when talking about “stretched
squarized Os” with respect to “perfect squarized Os”,
while other children seemed to only differentiate “perfect squarized Os” from all other squarized Os, since
they were special, being “all equal”.
The shared meanings
We chose to build on what seemed to be the idea of
this second group of students to reach a summary
of the shared meanings. The most important step in
this direction was a poster of “our” discoveries, a first
step towards the development of “mathematical texts”.
In this poster (Figure 6) several signs produced in the
classroom are reconsidered, constructing a text where
artifact signs (e.g. the figure of the bee-bot, the recollection of “giving it” a sequence of commands, the turns),
pivot signs (e.g. the squarized Os, the small arrow to
represent the external angles, and mathematical signs
(e.g. squares; numbers; rectangles) are included.
Figure 6: Poster of “our” discoveries
Is this text a mathematical text? Not yet: it is still a
hybrid text, where the richness of the exploration remains present. What is important in this phase is that
all of the students could identify this poster as having
been produced by the whole class as a community. The
choice of which signs to include was discussed by the
research team, trying to collect signs that hinted at the
individual and collective processes. The poster was
discussed in the classroom; the students seemed very
happy to find their ideas made public and to receive
a reduced-size copy to glue on their notebooks. Some
months later, a follow up questionnaire confirmed
that (at least some) students had appropriated, and
transferred it to a mathematical context, an inclusive
definition of rectangles (other students were still on
their way along this process). As mentioned before,
the process is not to be considered finished. The teacher has planned to go on with the same group of students and deepen the inclusive definition for which
she planted the seeds during this teaching experiment
in the first grade.
DISCUSSION
The teaching experiment fruitfully exploited the semiotic potential of the bee-bot, joining different ways
of representing the paths traced by the small
robot, as sequences of commands, as wholes,
as either physical or mental drawings, in
both dynamic and static ways. During this
long term process the students approached
several pieces of mathematics knowledge,
including counting (the commands), measuring (the length of the path, the distance),
exploring space, constructing and changing frames of reference, coordinating spatial perspectives, programming, planning
and debugging. The approach towards an
inclusive definition of rectangles is only one
aspect of this long and complex process.
A final comment on language. We do not
claim that the inclusive (and decontextualized) definition of rectangles is already
accepted by all the students (in fact we saw
that this was not the case). Rather we find it
important that students started becoming
aware of the fact that theoretical mathematical needs may be different from everyday
life needs. Moreover, we do not believe that
the inclusive definition should be used also
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Using pivot signs to reach an inclusive definition of rectangles and squares (Maria G. Bartolini Bussi and Anna Baccaglini-Frank)
in everyday life. Rather it seems that, with this experiment, we have put the students in the situation
of potentially seeing squares and rectangles within a
same “family”. What happened, indeed, was that the
idea of “square” seemed to be overarching, in spite
of the mathematical choices. The students seem to
speak of the squarized O as the ancestor of rectangles
(including squares) but, from the perceptual point of
view they need to distinguish “perfect squares” from
“stretched squares”. This reminds us of the Chinese
way of naming squares and rectangles: the sequences
of ideograms for the words “square” and rectangle”
contain two out of three of the same ideograms. Those
that indicate “sides” and “shape” are the same, while
the first indicates “exact” (for the square) and “long”
(for the rectangle). This is represented in Figure 7.
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in the mathematics classroom: Artifacts and signs after a
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of International Research in Mathematics Education, second edition (pp. 746–783). New York, NY: Routledge.
Bartolini Bussi, M.G., & Baccaglini-Frank, A. (2015). Geometry
in early years: sowing the seeds towards a mathematical
definition of squares and rectangles. ZDM Mathematics
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Clements, D. H. (2004). Geometric and Spatial Thinking in Early
Childhood Education. In D. H. Clements, & J. Sarama (Eds.),
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relations of quadrilaterals and prototype phenomenon.
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Hershkowitz, R. (1990). Psychological Aspects of Learning
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Figure 7: Chinese characters for “square” and “rectangle”,
respectively
University Press.
Koleza, E., & Giannisi, P. (2013). Kindergarten children’s reasoning about basic geometric shapes. In B. Ubuz, Ç. Haser, &
So, linguistically, a square is seen as a “shape with
exact sides” and a rectangle as a “(same) shape with
long sides”. In this case language makes explicit that
squares and rectangles are two kinds of a same thing,
deeply related to each other and not partitioned into
categories. The Chinese choice of the square as the
most important shape may be related to the Chinese
ancient culture, where it represents the Earth and
the circle represents the Sky. This Iconic cosmology
is shared by other ancient cultures.
M. A. Mariotti (Eds.), Proceedings of the Eighth Congress
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Education (pp. 2118–2127). Ankara, Turkey: Middle East
Technical University and ERME.
LOGO Foundation (2000). A LOGO primer or what’s with the
turtles? http://el.media.mit.edu/logofoundation/logo/turtle.
html. Accessed 31 Jan 2014.
Mariotti M. A., & Fischbein, E. (1997). Defining in classroom activities. Educational Studies in Mathematics, 34, 219–248.
ACKNOWLEDGEMENT
This teaching experiment was carried out within the
PerContare project (ASPHI, 2011), with the support of
the Fondazione per la Scuola of the Compagnia di San
Paolo di Torino.
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